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Rectified Schrödinger Bridge Matching for Few-Step Visual Navigation
Wuyang Luan, Junhui Li, Weiguang Zhao, Wenjian Zhang, Tieru Wu, Rui Ma
Intelligence
Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 94%
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Summary
Rectified Schrödinger Bridge Matching (RSBM) is a framework for few-step visual navigation in Embodied AI. It introduces an entropic regularization parameter ε to interpolate between standard Schrödinger Bridges and deterministic Optimal Transport. The authors prove that the conditional velocity field is invariant across the ε-spectrum and that reducing ε linearly decreases velocity variance, enabling high-fidelity trajectory generation in as few as 3 ODE steps without distillation.
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RSBM → uses → entropic regularization parameter ε
confidence 98% · controlled by a single entropic regularization parameter ε
Velocity Structure Invariance → enables → shared velocity-network parameterization
confidence 95% · enabling a single network to serve all regularization strengths
RSBM → improves → Visual Navigation
confidence 95% · RSBM achieves over 94% cosine similarity and 92% success rate in merely 3 integration steps
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Abstract
Abstract:Visual navigation is a core challenge in Embodied AI, requiring autonomous agents to translate high-dimensional sensory observations into continuous, long-horizon action trajectories. While generative policies based on diffusion models and Schrödinger Bridges (SB) effectively capture multimodal action distributions, they require dozens of integration steps due to high-variance stochastic transport, posing a critical barrier for real-time robotic control. We propose Rectified Schrödinger Bridge Matching (RSBM), a framework that exploits a shared velocity-field structure between standard Schrödinger Bridges ($\varepsilon=1$, maximum-entropy transport) and deterministic Optimal Transport ($\varepsilon\to 0$, as in Conditional Flow Matching), controlled by a single entropic regularization parameter $\varepsilon$. We prove two key results: (1) the conditional velocity field's functional form is invariant across the entire $\varepsilon$-spectrum (Velocity Structure Invariance), enabling a single network to serve all regularization strengths; and (2) reducing $\varepsilon$ linearly decreases the conditional velocity variance, enabling more stable coarse-step ODE integration. Anchored to a learned conditional prior that shortens transport distance, RSBM operates at an intermediate $\varepsilon$ that balances multimodal coverage and path straightness. Empirically, while standard bridges require $\geq 10$ steps to converge, RSBM achieves over 94% cosine similarity and 92% success rate in merely 3 integration steps -- without distillation or multi-stage training -- substantially narrowing the gap between high-fidelity generative policies and the low-latency demands of Embodied AI.
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- Source: https://arxiv.org/abs/2604.05673v1
- Canonical: https://arxiv.org/abs/2604.05673v1
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Rectified Schrödinger Bridge Matching for Few-Step Visual Navigation Wuyang Luan School of Mathematics Jilin University luanwy25@mails.jlu.edu.cn &Junhui Li11footnotemark: 1 College of Computer Science Chongqing University junhuili@stu.cqu.edu.cn &Weiguang Zhao Department of Computer Science University of Liverpool weiguang.zhao@liverpool.ac.uk &Wenjian Zhang GenY zhangwenjian@genycc.cn &Tieru Wu School of Mathematics Jilin University wutr@jlu.edu.cn &Rui Ma School of Artificial Intelligence Jilin University ruim@jlu.edu.cn Equal contribution.Corresponding author. Abstract Visual navigation is a core challenge in Embodied AI, requiring autonomous agents to translate high-dimensional sensory observations into continuous, long-horizon action trajectories. While generative policies based on diffusion models and Schrödinger Bridges (SB) effectively capture multimodal action distributions, they require dozens of integration steps due to high-variance stochastic transport, posing a critical barrier for real-time robotic control. We propose Rectified Schrödinger Bridge Matching (RSBM), a framework that exploits a shared velocity-field structure between standard Schrödinger Bridges (ε=1 \!=\!1, maximum-entropy transport) and deterministic Optimal Transport (ε→0 \!→\!0, as in Conditional Flow Matching), controlled by a single entropic regularization parameter ε . We prove two key results: (1) the conditional velocity field’s functional form is invariant across the entire ε -spectrum (Velocity Structure Invariance), enabling a single network to serve all regularization strengths; and (2) reducing ε linearly decreases the conditional velocity variance, enabling more stable coarse-step ODE integration. Anchored to a learned conditional prior that shortens transport distance, RSBM operates at an intermediate ε that balances multimodal coverage and path straightness. Empirically, while standard bridges require ≥10≥\!10 steps to converge, RSBM achieves over 94% cosine similarity and 92% success rate in merely 3 integration steps—without distillation or multi-stage training—substantially narrowing the gap between high-fidelity generative policies and the low-latency demands of Embodied AI. Code: https://github.com/WuyangLuan/RSBM. 1 Introduction Embodied Artificial Intelligence (Embodied AI) aims to develop autonomous agents capable of perceiving complex, dynamic environments and executing precise physical tasks. Visual navigation lies at the foundation of this objective, where an agent must continuously map streaming, high-dimensional sensory inputs into reliable action trajectories. Recently, generative policies powered by Denoising Diffusion Probabilistic Models (DDPMs) have shown remarkable success in Embodied AI, effectively capturing multimodal action distributions and mitigating the mode-collapse issues prevalent in traditional behavioral cloning. Despite these representational advantages, deploying standard diffusion-based policies on real-world robotic platforms presents a significant practical challenge: inference latency. Traditional diffusion models and standard Schrödinger Bridges (SB) rely on high-variance Brownian trajectories to map noise to data. Consequently, simulating these processes requires computationally intensive Stochastic Differential Equation (SDE) solvers and numerous iterative denoising steps. For an embodied agent with limited on-board compute and strict real-time control requirements, such delays are prohibitive. While accelerated sampling techniques exist, they often compromise the structural integrity of the generated action sequences, leading to suboptimal navigation performance. To harmonize generative capability with the low-latency demands of Embodied AI, we propose Rectified Schrödinger Bridge Matching (RSBM). Our key insight is that standard Schrödinger Bridges (ε=1 =1) and the linear interpolants underlying Conditional Flow Matching (ε→0 → 0) are not competing frameworks but endpoints of the same entropic regularization spectrum. By introducing an explicit regularization parameter ε∈(0,1] ∈(0,1] into the bridge transition kernel, RSBM provides principled interpolation between these extremes. We prove that the conditional velocity field’s functional form is invariant across the entire ε -family (Theorem 1), meaning a shared velocity-network parameterization applies across regularization strengths. At the same time, ε modulates the distribution of training samples, and we prove that reducing ε linearly reduces velocity variance (Proposition 1). Combined with a learned conditional prior that shortens the effective transport distance, RSBM enables high-fidelity trajectory generation in as few as 3 ODE steps. In summary, our main contributions are: • A Continuous SB–FM Interpolation Viewpoint: We show that ε -parameterized bridge kernels form a continuum from maximum-entropy Schrödinger Bridges (ε=1 \!=\!1) to deterministic optimal transport (ε→0 \!→\!0, equivalent to prior-conditioned CFM). We prove that the conditional velocity field is structurally invariant across this spectrum (Theorem 1: ε cancels exactly in dlogσε,t/dtd _ ,t/dt), enabling a shared velocity-network parameterization across the family. • Variance Reduction with Theoretical Characterization: We prove that ε -rectification linearly reduces conditional velocity variance (Proposition 1). This provides a direct mechanism consistent with the improved few-step integration behavior observed empirically. Full derivations from Schrödinger Bridge theory are provided in the appendix. • Single-Stage Few-Step Performance: RSBM achieves 6.3×6.3× lower MSE than NaviBridger at k=3k\!=\!3, and matches NaviBridger’s k=10k\!=\!10 accuracy with 3.8×3.8× fewer function evaluations across diverse navigation environments, without requiring distillation or multi-stage training—in contrast to Consistency Models Song et al. (2023) or Rectified Flow Liu et al. (2023) that need iterative reflow or teacher distillation. 2 Related Work Generative Policies for Visual Navigation. Visual navigation has evolved from classical modular pipelines Oriolo et al. (1995); Chalvatzaras et al. (2022); Yang et al. (2016) and end-to-end approaches based on reinforcement learning Zeng et al. (2020); Kulhánek et al. (2021) or behavioral cloning Chen et al. (2019); Manderson et al. (2020) toward generative policies. Foundation models such as ViNT Shah et al. (2023) demonstrate strong sample efficiency, while 3D representations like Gaussian Splatting Guo et al. (2025); Lei et al. (2025) improve spatial grounding. However, deterministic planners struggle in multi-modal environments where averaging across distinct valid modes yields infeasible plans Florence et al. (2022); Shafiullah et al. (2022). Score-based diffusion and flow matching models Carvalho et al. (2023); Ke et al. (2024); Zhu et al. (2023); Ajay et al. (2022); Janner et al. (2022) address this by capturing multi-modal action distributions; NoMaD Sridhar et al. (2024) was among the first to apply diffusion to navigation. A key limitation remains: initialization from isotropic Gaussian noise Ho et al. (2020); Sohl-Dickstein et al. (2015) necessitates long reverse processes, limiting real-time deployment Chen et al. (2016); Park et al. (2024). Figure 1: Denoising progression on two toy trajectories (star patrol and figure-8 loop). At t=Tt\!=\!T all methods start as unstructured point clouds (Gaussian noise for DDPM/FM; learned prior for NaviBridger/RSBM). By k=2k\!=\!2 (NFE==3), only RSBM closely matches the GT, while baselines require k≥10k\!≥\!10 to converge. At k=30k\!=\!30 (NFE==59), all methods have converged, confirming the quantitative trends in Table 3. Diffusion Bridges and Flow Matching. Diffusion bridges Somnath et al. (2023) generalize endpoint-conditioned processes Heng et al. (2025) via Doob’s h-transform Liu et al. (2022), with extensions to discrete-time Li et al. (2023) and continuous-time formulations Zhou et al. (2023). Recent Schrödinger Bridge Matching methods Shi et al. (2023); Liu et al. (2024); peluchetti2024bm2 learn bridges without iterative simulation; Light and Optimal SBM Gushchin et al. (2024a) and Adversarial SBM Gushchin et al. (2024b) further improve training efficiency, while Feedback SBM theodoropoulos2025feedback incorporates closed-loop supervision and Branched SBM Tang et al. (2025) handles multimodal transport topologies. In navigation, NaviBridger Ren et al. (2025) initializes a diffusion bridge from a learned motion prior, NaviD Zhang et al. (2024) leverages depth constraints, FlowNav Gode et al. (2025) combines CFM with depth priors, and StepNav Luo et al. (2026) constructs geometry-aware priors online. However, standard bridge formulations exhibit ill-conditioned dynamics near temporal boundaries, introducing truncation errors in few-step regimes Tong et al. (2023); Zhu and Lin (2024). Our method addresses this by explicitly constraining bridge variance via a single ε parameter, yielding more stable transport dynamics. Path Straightening and Accelerated Sampling. Rectified Flow Liu et al. (2023) straightens ODE trajectories via iterative reflow; Consistency Models Song et al. (2023) distill pre-trained diffusion models into few-step generators. Both require multi-stage training. RSBM achieves path straightening in a single stage by explicit variance control from an informed prior, retaining multi-modal coverage at intermediate ε . 3 Method Figure 2: Overview of the RSBM framework. Left: A dual-stream EfficientNet-B0 vision encoder fϕf_φ (§3.1) extracts observation and goal features, which are fused via positional encoding and self-attention into a context vector ∈ℝ256c ^256. Center: A learned variational prior network gψg_ψ (§3.1) produces a coarse action prior Ta_T. Right: A conditional U-Net 1D velocity network θv_θ (§3.3) with FiLM conditioning iteratively refines Ta_T into the output trajectory 0a_0 via an ε -Rectified Schrödinger Bridge (§3.2–3.4). The three trajectory snapshots illustrate the progressive refinement: from scattered prior waypoints (green), through partially converged intermediate states (orange), to the final high-fidelity trajectory (red) closely matching the ground truth (dashed gray). Our framework, illustrated in Figure 2, consists of three tightly coupled components. A dual-stream vision encoder fϕf_φ compresses streaming observations and the goal image into a context vector c (§3.1). A learned variational prior network gψg_ψ then maps c to a coarse action initialization Ta_T (§3.1). Finally, a conditional velocity network θv_θ drives an ε -rectified Schrödinger Bridge that progressively refines Ta_T into the output trajectory 0a_0 in as few as 3 ODE steps (§3.2–3.4). 3.1 Problem Formulation for Visual Navigation We formulate visual navigation as a conditional generative modeling problem. At each decision step, the agent receives a streaming observation sequence =It−C,…,ItO=\I_t-C,…,I_t\ of monocular RGB images and a goal image IgI_g, and must produce an action trajectory 0∈ℝH×2a_0 ^H× 2 representing H future waypoints in local coordinates. A perception encoder maps visual inputs into a compact context vector: =fϕ(,Ig)∈ℝd =f_φ(O,I_g) ^d (1) Rather than generating 0a_0 from uninformative Gaussian noise—a scheme that typically requires many denoising steps—we anchor the generative process to an informative conditional prior. A learned variational prior network produces a structured initialization Ta_T that coarsely captures navigational intent: T=gψ(,),∼qψ(∣,0)(train)/(,)(test) _T=g_ψ(z,c), q_ψ(z ,a_0)\;(train)\;/\;N(0,I)\;(test) (2) This prior Ta_T serves as the terminal boundary condition (t=Tt=T) for our diffusion bridge: the model need only learn the residual refinement from a coarse prior to the precise ground-truth trajectory, substantially shortening the effective transport distance. The generative task then reduces to constructing an efficient bridge from Ta_T to 0a_0. Figure 2 illustrates the complete RSBM pipeline. 3.2 The ε -Rectified Conditional Bridge Kernel Standard Diffusion Bridge Models construct a Brownian Bridge between 0a_0 and Ta_T with variance σt2=t2(1−t2/σmax2) _t^2=t^2(1-t^2/ _ ^2). These high-variance stochastic paths entangle intermediate states and necessitate many integration steps for accurate recovery. To rectify this, we introduce a structural regularization parameter ε∈(0,1] ∈(0,1] that explicitly controls the path geometry. The forward transition kernel of our rectified bridge is: qε(t∣0,T) q_ (a_t _0,a_T) =(t,σε,t2) =N\! ( μ_t,\; _ ,t^2\,I ) (3) where the mean seamlessly interpolates between data and prior: t μ_t =stT+(1−st)0,st=t2σmax2 =s_t\,a_T+(1-s_t)\,a_0, s_t= t^2 _ ^2 (4) and the variance is scaled by ε to enforce trajectory straightness: σε,t2 _ ,t^2 =ε⋅t2(1−st) = · t^2\! (1-s_t ) (5) Boundary conditions. For any ε∈(0,1] ∈(0,1], we have σε,02=0 _ ,0^2=0 (exact data recovery) and σε,σmax2=0 _ , _ ^2=0 (exact prior matching), ensuring the bridge correctly pins both endpoints. Geometric interpretation. Setting ε=1 =1 recovers the standard Brownian Bridge. As ε→0 → 0, the kernel collapses to δ(t−t)δ(a_t- μ_t)—the deterministic displacement interpolant of Monge–Kantarovich optimal transport. Intermediate values smoothly trade off between stochastic diversity and path straightness, providing a principled tuning knob for the navigation task. This formulation is grounded in entropic optimal transport and Schrödinger Bridge theory; we provide the complete derivation from stochastic optimal control principles in Appendix A. 3.3 Conditional Velocity Field and Flow Matching Objective To enable scalable training, we derive the conditional velocity field of the rectified bridge and formulate a simulation-free Flow Matching objective. Using the reparameterization t=t+σε,tϵa_t= μ_t+ _ ,t\, ε with ϵ∼(,) ε (0,I), the target velocity is: t∗(t∣0,T) _t^*(a_t _0,a_T) =dtdt+dlogσε,tdt(t−t) = d μ_tdt+ d _ ,tdt\,(a_t- μ_t) (6) Computing the key derivatives (complete algebra in Appendix B.1): dtdt d μ_tdt =2tσmax2(T−0) = 2t _ ^2\,(a_T-a_0) (7) dlogσε,tdt d _ ,tdt =1−2stt(1−st) = 1-2s_tt\,(1-s_t) (8) Theorem 1 (Velocity Structure Invariance). For the ε -rectified bridge family qε∈(0,1]\q_ \_ ∈(0,1] (Eq. 3), the logarithmic derivative of the standard deviation satisfies dlogσε,t/dt=(1−2st)/[t(1−st)]d _ ,t/dt=(1-2s_t)/[t\,(1-s_t)], which is independent of ε . Consequently, the functional form of the conditional velocity t∗v_t^* (Eq. 6) is invariant across the entire ε -spectrum: a single velocity network parameterization is valid for all ε∈(0,1] ∈(0,1]. Proof. Since σε,t=εt1−st _ ,t= \,t 1-s_t, the derivative is dσε,t/dt=ε(1−2st)/1−std _ ,t/dt= \,(1-2s_t)/ 1-s_t. Their ratio dσε,t/dtσε,t=ε(1−2st)/1−stεt1−st=1−2stt(1−st) d _ ,t/dt _ ,t= \,(1-2s_t)/ 1-s_t \,t 1-s_t= 1-2s_tt(1-s_t), where ε cancels exactly. Full derivation in Appendix B.1. ∎ Theorem 1 shows that SB (ε=1 \!=\!1) and FM (ε→0 \!→\!0) share the same conditional velocity functional form: ε controls the support of training samples ta_t via σε,t _ ,t (Eq. 5), while leaving the velocity field’s structure intact. In practice, ε acts as a spatial support constrictor—concentrating training samples near the deterministic interpolant t μ_t—yielding a geometrically simpler, lower-variance learning problem when ε<1 <1, as quantified in Proposition 1. We parameterize a neural velocity network θv_θ, implemented as a Conditional U-Net 1D with FiLM conditioning on c, and train via the simulation-free Conditional Flow Matching loss: ℒRSBM=t∼(0,T),0,T,ϵ[‖θ(t,t,)−t∗‖2] _RSBM=E_t (0,T),\,a_0,\,a_T,\, ε\! [ \|v_θ(a_t,t,c)-v_t^* \|^2 ] (9) Crucially, v-prediction directly parameterizes the ODE velocity field, avoiding the score-to-drift conversion required by x0x_0-prediction approaches. We empirically show this yields 35.6% lower error at k=3k=3; see Section 4.5. 3.4 Few-Step ODE Inference and Error Analysis At inference, action trajectories are generated by solving the Probability Flow ODE from Ta_T to 0a_0: dt=θ(t,t,)dt da_t=v_θ(a_t,t,c)\,dt (10) using a second-order Heun solver over a Karras timestep schedule t0>t1>⋯>tk=0\t_0>t_1>·s>t_k=0\: 1 _1 =θ(ti,ti,),~ti+1=ti+1(ti+1−ti) =v_θ(a_t_i,t_i,c), a_t_i+1=a_t_i+d_1\,(t_i+1-t_i) (11) 2 _2 =θ(~ti+1,ti+1,),ti+1=ti+12(1+2)(ti+1−ti) =v_θ( a_t_i+1,t_i+1,c), _t_i+1=a_t_i+ 12(d_1+d_2)\,(t_i+1-t_i) (12) Each step requires two function evaluations; the final evaluation of one step is reused as the first evaluation of the next (FSAL), yielding NFE=2k−1NFE=2k-1 for k steps. Why does ε<1 <1 enable fewer steps? We formalize this via the following result. Proposition 1 (Velocity Variance Reduction). For the ε -rectified bridge kernel qεq_ (Eq. 3), the conditional variance of the target velocity field satisfies: Var[t∗∣0,T]=ε⋅(1−2st)21−st⋅D \! [v_t^* _0,a_T ]= · (1-2s_t)^21-s_t·I_D (13) where D=2HD=2H denotes the flattened trajectory dimension. In particular, reducing ε linearly reduces the stochastic variation of the velocity field over the training distribution. Proof. From Eq. (6), the stochastic component of t∗v_t^* is dlogσε,tdt⋅σε,tϵ d _ ,tdt· _ ,t\, ε, where ϵ∼(,) ε (0,I). The per-component variance is therefore (dlogσε,tdt)2⋅σε,t2 ( d _ ,tdt )^2\!·\, _ ,t^2. Substituting Eqs. (5) and (8): (1−2st)2t2(1−st)2⋅ε⋅t2(1−st)=ε⋅(1−2st)21−st (1-2s_t)^2t^2(1-s_t)^2· · t^2(1-s_t)= · (1-2s_t)^21-s_t. ∎ Proposition 1 has two practical implications. First, lower velocity variance means the training target varies less across the bridge support, enabling the network to achieve better approximation quality with the same capacity. Second, in practice, smaller ε keeps trajectories closer to the interpolant t μ_t, producing lower-curvature paths that are easier to integrate with finite-step solvers. Appendix B.3 provides the corresponding sampling-error decomposition and its direct connection to ε . 4 Experiments 4.1 Experimental Setup Datasets & Simulation. We evaluate on five public navigation datasets: HuRoN, Recon, SACSoN, SCAND, and GoStanford, using the standard train/test splits from ViNT Shah et al. (2023) and NoMaD Sridhar et al. (2024) (total ∼ 60k trajectories across all five). Additionally, we construct a Gazebo-based Custom Indoor environment (500 train / 100 test episodes) and an outdoor CitySim environment (400 / 80). Episodes spawn with randomized start–goal pairs at 15–30 m separation. The robot receives monocular RGB images (96×9696× 96) and predicts 8-step waypoint trajectories. The regularization parameter ε=0.5 \!=\!0.5 was selected on Custom Indoor validation and held fixed for all other environments. Baselines & Metrics. We benchmark RSBM against ViNT (single-shot), NoMaD (k=20k=20), DDPM (k=50k=50), Conditional Flow Matching (CFM) (k=10k=10; Gaussian-to-data with linear interpolant, following Lipman et al. Lipman et al. (2023)), and NaviBridger Ren et al. (2025) (k=10k=10; standard Brownian Bridge with learned prior, ε=1 =1), evaluating Action MSE↓ , CosSim↑ , FDE↓ , Collision Rate↓ , and Success Rate↑ . All generative baselines use the same prior initialization Ta_T from gψg_ψ when applicable—specifically NaviBridger and RSBM—or standard Gaussian noise for DDPM and CFM, to isolate the effect of the generative dynamics. Implementation Details. RSBM employs an EfficientNet-B0 and Transformer-based vision encoder. Models are trained for 30 epochs using AdamW with learning rate 1×10−41× 10^-4 and batch size 256. The diffusion bridge utilizes a Heun 2nd-order ODE solver with boundary noise levels σmax=10.0 _ =10.0, σmin=0.002 _ =0.002, and our proposed spatial regularization ε=0.5 =0.5. Simulations run on Gazebo 11 and ROS 2 using a differential-drive TurtleBot3. Full architecture and hyperparameter details are provided in Appendix C. Reproducibility & Evaluation Protocol. All methods are trained on the same data splits and evaluated under the same observation horizon, waypoint format, and hardware—a single NVIDIA RTX 4090. Each experiment is repeated over 3 random seeds; we report the mean, with standard deviations provided in Appendix C.3. To reveal how each method degrades under tight compute budgets, we evaluate every generative baseline at both its default sampling budget and at k=3k\!=\!3 by reducing the solver steps at test time without retraining. We emphasize that this zero-shot step-count reduction is applied uniformly to all methods, including RSBM. Unlike distillation-based accelerators such as Consistency Models Song et al. (2023), RSBM requires no additional training stages for few-step inference—the same model trained once is evaluated at arbitrary k. 4.2 Main Results Table 1 presents a comprehensive comparison across all methods. Each algorithm is shown at both its default sampling budget and at k=3k\!=\!3, revealing how gracefully each approach degrades under zero-shot step reduction. RSBM at k=3k\!=\!3 (NFE=\,=\,5) not only matches but often surpasses baselines at their full default budgets; increasing RSBM to k=10k\!=\!10 yields only marginal further gains (MSE 1.90→1.721.90\!→\!1.72, Suc. 92→93%92\!→\!93\%), confirming early saturation. In Custom Indoor, RSBM achieves a 92% success rate and 0.945 CosSim with only 5 NFEs—a +4% success rate improvement over NaviBridger at k=10k\!=\!10 (NFE=\,=\,19), using 3.8×3.8× fewer function evaluations. Wall-clock latency measurements are provided in Appendix C.7. Meanwhile, other methods degrade significantly under this zero-shot step reduction: NaviBridger’s CosSim drops from 0.942 to 0.710, and DDPM falls to 0.320. Table 1: Comprehensive comparison. Each method is shown at its default k and at k=3k\!=\!3, revealing degradation under zero-shot step reduction. Gray rows: k=3k\!=\!3 variants. RSBM is shown at both k=3k\!=\!3 and k=10k\!=\!10 to demonstrate early saturation. Bold: best overall per column. Custom Indoor CitySim (Outdoor) Method k NFE MSE↓ CosSim↑ FDE↓ Col.%↓ Suc.%↑ MSE↓ CosSim↑ FDE↓ Col.%↓ Suc.%↑ ViNT Shah et al. (2023) (1-shot) 1 1 6.50 0.720 2.85 1.58 28 8.20 0.650 4.50 0.41 38 NoMaD Sridhar et al. (2024) 20 20 3.60 0.820 1.95 1.32 32 5.80 0.740 3.20 0.34 52 NoMaD Sridhar et al. (2024) 3 6 8.40 0.610 3.75 2.25 18 10.50 0.540 5.20 0.55 22 DDPM Ho et al. (2020) 50 50 3.80 0.820 2.05 0.98 64 5.50 0.750 3.10 0.35 50 DDPM Ho et al. (2020) 3 6 14.80 0.320 6.10 3.60 6 16.20 0.280 7.80 0.72 4 FM Lipman et al. (2023) 10 10 2.80 0.910 1.45 0.52 82 4.20 0.850 2.20 0.32 58 FM Lipman et al. (2023) 3 3 5.90 0.710 2.90 1.15 45 7.80 0.650 3.80 0.48 34 NaviBridger Ren et al. (2025) 10 19 1.82 0.942 0.82 0.41 88 2.50 0.920 1.15 0.30 64 NaviBridger Ren et al. (2025) 3 5 12.00 0.710 4.20 2.80 35 13.50 0.660 5.60 0.65 28 RSBM (Ours) 3 5 1.90 0.945 0.80 0.38 92 2.55 0.925 1.10 0.28 68 RSBM (Ours) 10 19 1.72 0.949 0.75 0.35 93 2.40 0.930 1.05 0.26 70 Figure 4 visualizes the quality–cost Pareto frontier by plotting CosSim and Success Rate against NFE for every method. RSBM at k=3k\!=\!3 (NFE=5NFE=5) lies in the upper-left region in both panels, achieving 0.945 CosSim and 92% success rate—comparable to NaviBridger at k=10k\!=\!10 (NFE=19NFE=19) while using 3.8×3.8× fewer evaluations. The dashed lines connecting each method’s low-k and high-k configurations show performance degradation under budget constraints: DDPM and NaviBridger degrade sharply, whereas RSBM maintains strong performance at minimal cost. This trend is consistent with Proposition 1: restricting spatial support via ε<1 <1 reduces velocity variance, which aligns with smoother few-step ODE integration in practice. 4.3 Ablation: ε Regularization Figure 4.3 dissects the regularization parameter ε . Setting ε=1.0 =1.0 recovers standard SB, yielding highly curved paths that degrade at low k. Decreasing ε progressively straightens these paths. Notably, very small values (ε∈0.1,0.3 ∈\0.1,0.3\) approach the deterministic OT interpolant (ε→0 → 0, equivalent to a prior-conditioned OT-CFM): they converge faster but exhibit over-regularization and degraded diversity, producing brittle trajectories at ambiguous intersections. We use ε=0.5 =0.5 as the default operating point: it retains sufficient stochasticity to capture multi-modal navigation intent while maintaining high-fidelity generation at k=3k=3. Overall, RSBM with ε∈(0,1) ∈(0,1) provides a favorable few-step tradeoff between the two extremes—full SB (ε=1 =1) and pure OT-CFM (ε→0 → 0). Figure 3: Effect of ε . ε=0.5 =0.5 (red) provides stable low-k performance. Disentangling prior and bridge contributions. Table 8 in Appendix C.6 reports a four-way decomposition (Gaussian+SB, Gaussian+RSBM, Prior+SB, Prior+RSBM). The learned prior reduces transport distance, lowering MSE from 12.012.0 to 5.85.8 (2.1×2.1×), while ε -rectification straightens ODE paths, further lowering MSE from 5.85.8 to 1.91.9 (3.1×3.1×). These gains are multiplicative—neither component alone approaches the full system. Moreover, ε -rectification from Gaussian noise already achieves 2.9×2.9× lower MSE than standard SB (4.24.2 vs. 12.012.0), confirming that bridge rectification contributes independently of prior quality. 4.4 Robustness Across Real-World Datasets To validate that the advantage observed in Custom Indoor generalizes, Table 2 breaks down Action MSE and CosSim across five diverse real-world datasets in the standard open-loop offline protocol of Shah et al. (2023); Sridhar et al. (2024). At k=3k\!=\!3, RSBM remains competitive with NaviBridger at k=10k\!=\!10 across all five datasets: NaviBridger has an average MSE of 4.424.42 and CosSim of 0.6720.672, while RSBM obtains 1.191.19 and 0.9340.934. The gap is most pronounced on GoStanford, a long-range outdoor dataset, and SACSoN, which features dynamic obstacles—domains where high path curvature amplifies few-step truncation error—consistent with the variance reduction mechanism of Proposition 1. Extended per-dataset multi-metric results including FDE are provided in Appendix C.4. Figure 4: Quality–cost Pareto frontier. Each marker represents a method at a given sampling budget (k). (a) CosSim vs. NFE; (b) Success Rate vs. NFE. RSBM at k=3k\!=\!3 (NFE=5\!=\!5) lies on the favorable frontier region, providing strong quality at substantially lower evaluations. Table 2: Per-dataset generalization. Action MSE↓ and CosSim↑ across five diverse real-world datasets. RSBM(k=3k\!=\!3) consistently matches or exceeds NaviBridger(k=10k\!=\!10); NaviBridger(k=3k\!=\!3) degrades significantly across all domains. Bold: best per column among k=3k\!=\!3 methods. Action MSE ↓ CosSim ↑ Method k HuRoN Recon SACSoN SCAND GoStan. Avg HuRoN Recon SACSoN SCAND GoStan. Avg DDPM Ho et al. (2020) 10 1.38 1.19 1.74 0.72 4.12 1.83 0.880 0.900 0.860 0.920 0.780 0.868 DDPM Ho et al. (2020) 3 8.50 6.20 9.80 4.60 15.30 8.88 0.350 0.420 0.280 0.480 0.210 0.348 FM Lipman et al. (2023) 10 1.10 0.97 1.55 0.65 3.65 1.58 0.910 0.920 0.885 0.935 0.820 0.894 FM Lipman et al. (2023) 3 3.20 2.60 4.10 1.80 7.50 3.84 0.720 0.740 0.670 0.790 0.560 0.696 NaviBridger Ren et al. (2025) 10 0.27 0.88 1.43 0.59 3.22 1.28 0.955 0.935 0.920 0.960 0.875 0.929 NaviBridger Ren et al. (2025) 3 2.80 3.45 5.20 2.15 8.50 4.42 0.750 0.690 0.620 0.780 0.520 0.672 RSBM 3 0.25 0.82 1.35 0.48 3.05 1.19 0.958 0.940 0.925 0.965 0.880 0.934 RSBM 10 0.24 0.80 1.32 0.47 2.95 1.16 0.960 0.942 0.928 0.968 0.885 0.937 4.5 Ablation: Prediction Target Table 3 compares three prediction targets under the same RSBM bridge (ε=0.5 =0.5, Custom Indoor). v-prediction directly parameterizes the ODE velocity field, avoiding the score-to-drift conversion of ϵε-prediction and the endpoint estimation bias of x0x_0-prediction. At k=3k\!=\!3, v-prediction achieves 35.6% lower MSE than x0x_0-prediction and 45.7% lower than ϵε-prediction. The gap narrows at k=10k\!=\!10, and by k=50k\!=\!50 all three targets converge (MSE 1.67 vs. 1.72 vs. 1.74), confirming that v-prediction’s advantage is concentrated in the few-step regime. Table 3: Prediction target ablation (RSBM, ε=0.5 \!=\!0.5, Custom Indoor). v-prediction’s advantage is most pronounced at low k. Bold: best per column. k=3k\!=\!3 (NFE=5) k=10k\!=\!10 (NFE=19) Target MSE↓ CosSim↑ FDE↓ MSE↓ CosSim↑ FDE↓ ϵε-pred 3.50 0.895 1.42 1.95 0.940 0.88 x0x_0-pred 2.95 0.920 1.15 1.80 0.946 0.84 v-pred (Ours) 1.90 0.945 0.80 1.72 0.949 0.78 4.6 Qualitative Results Figure 5 visualizes predicted trajectories across eight challenging scenarios arranged in a 2×42\!×\!4 grid, each featuring multi-turn paths with 3–7 forced direction changes. Under a tight k=3k\!=\!3 budget, baselines collide with obstacles within the first few turns (× markers), and their invalid continuations are shown as faint dotted lines. In contrast, RSBM produces smooth, collision-free trajectories closely tracing the ground truth even through zigzag corridors, spiral environments, and warehouse aisles. This spatial consistency is consistent with the variance reduction established in Proposition 1: the rectified bridge concentrates ODE trajectories near the interpolant t μ_t, producing clean, stable paths even with minimal integration steps. Figure 5: Qualitative trajectory comparison across eight challenging scenarios (2×42\!×\!4 grid, k=3k\!=\!3, NFE=5\!=\!5). Top row: four indoor/structured environments. Bottom row: four large-scale environments. Baselines collide early (×); faint dotted lines show invalid ghost continuations. RSBM (green) remains collision-free and closely tracks the ground truth (dashed gray). 4.7 Real-Robot Deployment To validate real-world applicability, we deploy all four methods on a quadruped robot (Alphababy, monocular 1280×7201280\!×\!720 RGB at 4 Hz, NVIDIA Jetson Orin) in a small number of indoor scenarios (∼40 40 s episodes). Two representative scenes are: (A) a corridor requiring a right 90∘90 turn, and (B) a furnished room requiring a left 90∘90 turn. RSBM (k=3k\!=\!3, ∼50 50 ms/cycle) completes both with smooth, collision-free trajectories. NaviBridger and FM also succeed but exhibit jitter at turning points. DDPM (k=50k\!=\!50, ∼350 350 ms/cycle) fails both scenes due to control-loop lag. Videos are in the supplementary material. 5 Conclusion We introduced Rectified Schrödinger Bridge Matching (RSBM), a framework that unifies Schrödinger Bridges and Flow Matching through a single entropic regularization parameter ε . We proved that the velocity field’s functional form is invariant across the entire ε -spectrum (Theorem 1) and that ε -rectification linearly reduces velocity variance (Proposition 1), enabling high-fidelity generation in as few as 3 ODE steps. Combined with a learned conditional prior, RSBM achieves 94.5% cosine similarity and 92% success rate at k=3k\!=\!3, matching NaviBridger’s full-budget accuracy with 3.8×3.8× fewer function evaluations—without distillation or multi-stage training. Unlike Consistency Models Song et al. (2023) or Rectified Flow Liu et al. (2023), which require additional training stages, RSBM achieves path straightening through bridge geometry in a single stage; these approaches are potentially complementary. The same policy runs at arbitrary step budgets without retraining, providing a direct latency–quality knob for deployment across heterogeneous platforms. Limitations. 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Appendix A Theoretical Foundations: From Stochastic Control to Rectified Bridges This appendix presents the full theoretical lineage connecting stochastic optimal control, Schrödinger Bridges, and our ε -rectified formulation. The main text (Section 3) presents only the operational formulas needed for training and inference; here we provide the complete derivation. A.1 Generative Modeling as Stochastic Optimal Control We cast trajectory generation as a Stochastic Optimal Control (SOC) problem. The goal is to find an optimal control utu_t steering a base distribution towards a structured prior while minimizing path cost: minu _u X∼pu[∫0112‖ut(Xt)‖2t+g(X1)] _X p^u [ _0^1 12\|u_t(X_t)\|^2dt+g(X_1) ] (14) s.t. dXt=(ft(Xt)+σtut(Xt))dt+σtdWt,X0∼pdata dX_t= (f_t(X_t)+ _tu_t(X_t) )dt+ _tdW_t, X_0 p_data (15) where g(X1)g(X_1) is the terminal cost. By the Hamilton-Jacobi-Bellman (HJB) equation, the optimal control and value function satisfy: ut⋆(x) u _t(x) =−σt∇Vt(x) =- _t∇ V_t(x) (16) Vt(x) V_t(x) =−logX∼pbase[exp(−g(X1))∣Xt=x] =- _X p_base[ (-g(X_1)) X_t=x] (17) p⋆(X0,X1) p (X_0,X_1) =pbase(X0,X1)exp(−g(X1)+V0(X0)) =p_base(X_0,X_1) (-g(X_1)+V_0(X_0)) (18) Computing p⋆(X0,X1)p (X_0,X_1) is intractable. Under a memoryless relaxation pbase0,1(X0,X1)≡pbase0(X0)pbase1(X1)p_base^0,1(X_0,X_1)≡ p_base^0(X_0)\,p_base^1(X_1), the initial value function flattens: V0(X0) V_0(X_0) ≡const,p∗(X1)∝pbase(X1)exp(−g(X1))≡pprior(X1) , p (X_1) p_base(X_1) (-g(X_1))≡ p_prior(X_1) (19) implying the terminal cost is g(x)=logpbase1(x)pprior(x)g(x)= p_base^1(x)p_prior(x). A.2 Duality with Schrödinger Bridges The Schrödinger Bridge (SB) framework provides rigorous optimal transport guarantees. Under SB optimality, dynamics are governed by coupled potentials ϕt _t and ϕ^t φ_t: ut⋆(x) u _t(x) =σt∇xlogϕt(x),vt⋆(x)=σt∇xlogϕ^t(x) = _t _x _t(x), v _t(x)= _t _x φ_t(x) (20) ϕt(x) _t(x) =∫pbase1|t(y∣x)ϕ1(y)y,ϕ0ϕ^0=pprior = p_base^1|t(y x)\, _1(y)\,dy, _0 φ_0=p_prior (21) ϕ^t(x) φ_t(x) =∫pbaset|0(x∣y)ϕ^0(y)y,ϕ1ϕ^1=pdata = p_base^t|0(x y)\, φ_0(y)\,dy, _1 φ_1=p_data (22) The SOC and SB formulations are unified via Hopf-Cole transforms: ϕt(x)=exp(−Vt(x)) _t(x)= (-V_t(x)) and ϕ^t(x)=exp(Vt(x))pt⋆(x) φ_t(x)= (V_t(x))\,p _t(x), yielding g(x)=logϕ^1(x)pprior(x)g(x)= φ_1(x)p_prior(x). A.3 Connection to the ε -Rectified Kernel In the standard SB with Variance Exploding (VE) dynamics dXt=σtdWtdX_t= _t\,dW_t, the conditional bridge kernel between pinned endpoints (X0,X1)(X_0,X_1) is: q(Xt∣X0,X1)=(Xt;(1−st)X0+stX1,t2(1−st)),st=t2σmax2 q(X_t X_0,X_1)=N\! (X_t;\;(1-s_t)\,X_0+s_t\,X_1,\;t^2(1-s_t)\,I ), s_t= t^2 _ ^2 (23) The conditional variance σt2=t2(1−st) _t^2=t^2(1-s_t) has two important properties: (i) it vanishes at the boundary (σ02=0 _0^2=0, σmax2=0 _ _ ^2=0), pinning the bridge to its endpoints; (i) it peaks at the midpoint tpeak=σmax/2t_peak= _ / 2, where stochastic transport is most diffuse. Our ε -rectification replaces this variance with: σε,t2=ε⋅t2(1−st),ε∈(0,1] _ ,t^2= · t^2(1-s_t), ∈(0,1] (24) This modulates the entropic regularization strength of the Schrödinger Bridge. Specifically, the SB problem with entropic cost γKL(p∥pref) (p\,\|\,p_ref) produces a family of solutions indexed by γ; our ε plays a role analogous to γ/γ0γ/ _0 where γ0 _0 is the reference regularization. This interpolates between two extremes: • ε=1 =1: Standard Brownian Bridge (maximum entropy, full stochastic transport). • ε→0 → 0: Deterministic displacement interpolant t μ_t (minimum entropy, Monge OT map). By choosing ε∈(0,1) ∈(0,1), we retain the bridge structure and boundary conditions while concentrating probability mass near the geodesic connecting 0a_0 and Ta_T. This provides a principled mechanism for trading off generation diversity against path straightness. A.3.1 Formal KL Divergence Connection We now rigorize the entropic regularization interpretation. Consider the Schrödinger Bridge problem with regularization strength γ>0γ>0: minp∈(Ω) _p ( ) ∫c(ω)p(ω)+γKL(p∥pref) c(ω)\,dp(ω)+γ\,KL(p\,\|\,p_ref) (25) where c(ω)=12∫0T‖ut‖2tc(ω)= 12 _0^T\|u_t\|^2\,dt is the kinetic energy cost, prefp_ref is the Brownian Bridge reference, and Ω is the path space. The solution satisfies p⋆∝pref⋅exp(−c/γ)p p_ref· (-c/γ), with conditional kernel: p⋆(t∣0,T)=(t,γ0t2(1−st)) p (a_t _0,a_T)=N\! ( μ_t,\; γ _0\,t^2(1-s_t)\,I ) (26) where γ0 _0 is the reference regularization yielding unit bridge variance. Identifying ε=γ/γ0 =γ/ _0, we obtain σε,t2=ε⋅t2(1−st) _ ,t^2= · t^2(1-s_t), recovering Eq. (24). The KL cost between the ε -rectified and standard bridges admits a closed form: KL(qε(t∣0,T)∥q1(t∣0,T))=D2(ε−1−logε) \! (q_ (a_t _0,a_T)\, \|\,q_1(a_t _0,a_T) )= D2 ( -1- ) (27) where D is the action space dimensionality. This is strictly non-negative for ε≠1 ≠ 1 and monotonically increases as ε→0 → 0, quantifying the information cost of rectification. For our default ε=0.5 =0.5 with D=16D=16 (8 waypoints × 2 dimensions), KL=8×(0.5−1−log0.5)=1.55KL=8×(0.5-1- 0.5)=1.55 nats—a moderate cost that substantially reduces transport curvature while preserving sufficient stochasticity for multi-modal generation. A.4 Boundary Condition Verification A critical requirement for any modified bridge kernel is that the boundary conditions remain exact. We verify this for the ε -rectified kernel: t=t+σε,tϵ,ϵ∼(,) _t= μ_t+ _ ,t\, ε, ε (0,I) (28) where t=(1−st)0+stT μ_t=(1-s_t)\,a_0+s_t\,a_T and σε,t=εt1−st _ ,t= \,t\, 1-s_t. At t=0t=0: s0=0,σε,0=ε⋅0⋅1=0⟹0=(1−0)0+0⋅T+0=0✓ s_0=0, _ ,0= · 0· 1=0 _0=(1-0)\,a_0+0·a_T+0=a_0\; (29) At t=σmaxt= _ : sσmax=σmax2σmax2=1,σε,σmax=ε⋅σmax⋅0=0⟹σmax=0⋅0+1⋅T=T✓ s_ _ = _ ^2 _ ^2=1, _ , _ = · _ · 0=0 _ _ =0·a_0+1·a_T=a_T\; (30) Thus, the ε -rectified bridge is exact at both endpoints for any ε>0 >0. The modification affects only the interior of the trajectory (0<t<σmax)(0<t< _ ), where it controls how much the sampled path deviates from the deterministic interpolant t μ_t. A.5 Probability Flow ODE Derivation Given the forward bridge SDE dt=t(t)dt+gtdtda_t=f_t(a_t)\,dt+g_t\,dW_t, the marginal-preserving Probability Flow ODE (PF-ODE) Song et al. [2020] replaces stochastic dynamics with a deterministic flow that produces identical marginal distributions pt(t)p_t(a_t) at every time t: dtdt=t(t)−12gt2∇tlogpt(t) da_tdt=f_t(a_t)- 12g_t^2\, _a_t p_t(a_t) (31) For our ε -rectified bridge with conditional kernel t∣(0,T)∼(t,σε,t2)a_t (a_0,a_T) ( μ_t, _ ,t^2\,I), the conditional score is: ∇tlogp(t∣0,T)=−t−tσε,t2 _a_t p(a_t _0,a_T)=- a_t- μ_t _ ,t^2 (32) Substituting into the PF-ODE and rearranging, we obtain the velocity field that the neural network θv_θ learns to approximate: t∗(t) _t^*(a_t) =dtdt+dlogσε,tdt(t−t) = d μ_tdt+ d _ ,tdt\,(a_t- μ_t) (33) The key observation is that dlogσε,t/dt=(1−2st)/[t(1−st)]d _ ,t/dt=(1-2s_t)/[t(1-s_t)] is independent of ε (as proved in Appendix B.1, Step 3). This means the same velocity functional form applies regardless of ε ; only the distribution of training inputs ta_t changes. During inference, we integrate the PF-ODE backward from t=σmaxt= _ to t→0t→ 0: t−Δt=t−Δt⋅θ(t,t,) _t- t=a_t- t·v_θ(a_t,t,c) (34) using Heun’s method (2nd-order Runge-Kutta) with k steps, yielding NFE= 2k−1\,=\,2k-1 function evaluations. A.6 v-Prediction: Signal-to-Noise Ratio Analysis The velocity field t∗v_t^* admits three equivalent reparameterizations—ε -prediction, x0x_0-prediction, and v-prediction—each corresponding to a different training target. We analyze why v-prediction yields superior performance, particularly at low step counts. ε -prediction. The network predicts the noise ϵ ε; the velocity is recovered via: t=dtdt+dσε,tdtϵ _t= d μ_tdt+ d _ ,tdt\, ε (35) The training signal is ϵ∼(,) ε (0,I), which is independent of t. Near t≈0t≈ 0, the coefficient dσε,t/dtd _ ,t/dt is small, so small errors in ϵ ε are amplified when reconstructing the trajectory. x0x_0-prediction. The network predicts the clean target ^0 a_0; the velocity is: t=dtdt+dlogσε,tdt(t−(1−st)^0−stT) _t= d μ_tdt+ d _ ,tdt\,(a_t-(1-s_t) a_0-s_t\,a_T) (36) Near t≈σmaxt≈ _ , the signal-to-noise ratio for ^0 a_0 is low since ta_t is dominated by the terminal point Ta_T. v-prediction (Ours). The network directly predicts the velocity ^t=t∗ v_t=v_t^*. This naturally balances the mean drift and stochastic terms across all t: SNRv(t)=‖dt/dt‖2Var[t∗∣0,T]=‖dt/dt‖2ε(1−2st)2/(1−st) _v(t)= \|d μ_t/dt\|^2Var[v_t^* _0,a_T]= \|d μ_t/dt\|^2 \,(1-2s_t)^2/(1-s_t) (37) This ratio is well-behaved across the full interval t∈(0,σmax)t∈(0, _ ), avoiding the boundary singularities of both ε -prediction (near t=0t=0) and x0x_0-prediction (near t=σmaxt= _ ). Furthermore, v-prediction directly minimizes the ODE integration error, since the Heun solver accumulates velocity predictions—making it the most natural parameterization for few-step inference. Appendix B Detailed Derivations and Proofs B.1 Full Velocity Field Derivation Starting from the reparameterized bridge sample t=t+σε,tϵa_t= μ_t+ _ ,t\, ε, ϵ∼(,) ε (0,I): Step 1: Mean derivative. dtdt=dstdt(T−0)=2tσmax2(T−0) d μ_tdt= ds_tdt\,(a_T-a_0)= 2t _ ^2\,(a_T-a_0) (38) Step 2: Standard deviation derivative. σε,t _ ,t =εt1−st = \,t\, 1-s_t (39) dσε,tdt d _ ,tdt =ε[1−st+t⋅−dst/dt21−st]=ε[1−st−st1−st] = [ 1-s_t+t· -ds_t/dt2 1-s_t ]= [ 1-s_t- s_t 1-s_t ] (40) =ε1−2st1−st = \, 1-2s_t 1-s_t (41) Step 3: Logarithmic derivative and ε cancellation. dlogσε,tdt=1σε,t⋅dσε,tdt=1εt1−st⋅ε1−2st1−st=1−2stt(1−st) d _ ,tdt= 1 _ ,t· d _ ,tdt= 1 \,t 1-s_t· \, 1-2s_t 1-s_t= 1-2s_tt(1-s_t) (42) The ε factors cancel exactly, proving that the functional form of dlogσε,t/dtd _ ,t/dt—and hence the deterministic drift structure of the target velocity (Eq. 6)—is ε -invariant. Note that ε still affects the distribution of training points ta_t via σε,t _ ,t, which modulates the stochastic component’s variance (Proposition 1). Step 4: Final velocity expression. Substituting ϵ=(t−t)/σε,t ε=(a_t- μ_t)/ _ ,t: t∗=2tσmax2(T−0)+1−2stt(1−st)(t−t) _t^*= 2t _ ^2\,(a_T-a_0)+ 1-2s_tt(1-s_t)\,(a_t- μ_t) (43) B.2 Discussion of Proposition 1 The complete proof is given in the main text (Section 3.4). Here we discuss its consequences in greater detail. Variance landscape. The function V(st)=ε⋅(1−2st)2/(1−st)V(s_t)= ·(1-2s_t)^2/(1-s_t) has the following structure: • It vanishes at st=1/2s_t=1/2, corresponding to t=σmax/2t= _ / 2, where the bridge midpoint concentrates all probability mass on the interpolant t μ_t, and the velocity becomes purely deterministic regardless of ε . • Near st→0s_t→ 0 (start of transport), V(st)→εV(s_t)→ , so the variance is bounded by ε . • Near st→1s_t→ 1 (end of transport), V(st)→+∞V(s_t)→+∞, reflecting the bridge’s pinching towards the terminal point Ta_T. This singularity is canceled by the vanishing time increment Δt→0 t→ 0 in the ODE solver, so the integrated contribution remains finite. Comparison with standard bridges. For the unrectified bridge (ε=1 =1), the variance V(st)=(1−2st)2/(1−st)V(s_t)=(1-2s_t)^2/(1-s_t) has unit-scale magnitude near the boundaries. Reducing ε to 0.50.5 halves this variance uniformly over t, which (i) reduces the difficulty of the regression problem for the neural network, and (i) makes the ODE right-hand-side smoother and hence better approximated by a finite-step Heun solver. This two-fold benefit explains why the MSE gap between standard SB and RSBM is most pronounced at low k (Table 1) and narrows as k increases (Table 3). B.3 Sampling Error Analysis We decompose the total sampling error into two sources: network approximation and ODE discretization. This factorization is standard in the flow matching literature Lipman et al. [2023]; we state it here for completeness. Standard error decomposition. Let δ2=‖θ−∗‖2δ^2=E\|v_θ-v^*\|^2 denote the functional approximation error, and assume (i) θv_θ is LθL_θ-Lipschitz on the relevant support, and (i) the ODE flow map is non-expansive. For a k-step Heun solver simulating the PF-ODE (10), the 2-Wasserstein distance satisfies: W2(p^0,p0)≤C1Tδ+C2LθT3k2 W_2( p_0,p_0)≤ C_1Tδ+ C_2L_θT^3k^2 (44) where C1,C2C_1,C_2 are absolute constants depending only on the time horizon and solver order. This bound holds for any velocity-parameterized ODE model, independent of ε . Direct connection to ε . By Proposition 1, the training target has conditional variance proportional to ε . In the error decomposition of Eq. (44), lowering target variance reduces regression difficulty and decreases discretization error in few-step integration. This theoretical direction is consistent with the empirical behavior in Figure 4.3, where smaller ε improves low-k stability until over-regularization appears at very small values. B.4 Training and Inference Algorithms We provide complete pseudocode for the RSBM training and inference procedures. Algorithm B.4 describes the training loop, and Algorithm B.4 describes the inference (sampling) procedure. Algorithm 1: RSBM Training Input: Dataset =(i,Igi,0i)i=1ND=\(O_i,I_g^i,a_0^i)\_i=1^N, noise schedule σmax _ , rectification parameter ε , learning rate η Output: Trained networks fϕf_φ (vision encoder), gψg_ψ (variational prior), θv_θ (velocity) repeat 1. Sample minibatch (,Ig,0)∼\(O,I_g,a_0)\ 2. Encode context: =fϕ(,Ig)c=f_φ(O,I_g) 3. Compute prior: T=gψ(,)a_T=g_ψ(c,z), ∼qψ(∣,0)\;z q_ψ(z ,a_0) 4. Sample time: t∼(σmin,σmax)t ( _ ,\; _ ) 5. Compute schedule: st=t2/σmax2s_t=t^2/ _ ^2 6. Compute interpolant: t=(1−st)0+stT μ_t=(1-s_t)\,a_0+s_t\,a_T 7. Sample noise: ϵ∼(,) ε (0,I) 8. Construct bridge sample: t=t+εt1−stϵa_t= μ_t+ \,t 1-s_t\; ε 9. Compute target velocity: t∗=2tσmax2(T−0)+1−2stt(1−st)(t−t)v_t^*= 2t _ ^2(a_T-a_0)+ 1-2s_tt(1-s_t)(a_t- μ_t) 10. Update: (ϕ,ψ,θ)←(ϕ,ψ,θ)−η∇[‖θ(t,t,)−t∗‖2+ℒprior](φ,ψ,θ)←(φ,ψ,θ)-η\,∇\! [\|v_θ(a_t,t,c)-v_t^*\|^2+L_prior ] until converged Algorithm 2: RSBM Inference (Sampling) Input: Observation O, goal image IgI_g, trained networks (fϕ,gψ,θ)(f_φ,g_ψ,v_θ), steps k, schedule tii=0k\t_i\_i=0^k Output: Predicted trajectory ^0 a_0 1. Encode context: =fϕ(,Ig)c=f_φ(O,I_g) 2. Generate prior: T=gψ(,prior)a_T=g_ψ(c,z_prior), prior∼(,)\;z_prior (0,I) 3. Initialize: t0←Ta_t_0 _T, where t0=σmaxt_0= _ 4. for i=0,1,…,k−1i=0,1,…,k-1 do Δt=ti−ti+1 t=t_i-t_i+1 // Heun’s method (2nd-order Runge-Kutta): 1=θ(ti,ti,)d_1=v_θ(a_t_i,\;t_i,\;c) [1st NFE] ~=ti−Δt⋅1 a=a_t_i- t·d_1 2=θ(~,ti+1,)d_2=v_θ( a,\;t_i+1,\;c) [2nd NFE] ti+1=ti−Δt⋅12(1+2)a_t_i+1=a_t_i- t· 12(d_1+d_2) 5. return ^0=tk a_0=a_t_k // Time schedule: Karras sigmas ti=(σmax1/ρ+ik(σmin1/ρ−σmax1/ρ))ρt_i= ( _ ^1/ρ+ ik( _ ^1/ρ- _ ^1/ρ) )^ρ, ρ=7ρ=7 // Total NFE =2k−1=2k-1 (last step omits the correction evaluation) Appendix C Implementation Details C.1 Network Architecture Table 4: Architecture and training hyperparameters. Component Specification Vision encoder EfficientNet-B0 + 4-layer Transformer Context dimension d 256 Prior encoder qψq_ψ 3-layer MLP, hidden dim 256 Prior decoder gψg_ψ 3-layer MLP, hidden dim 256 Latent dimension |||z| 32 Velocity prediction network Conditional U-Net 1D U-Net channels [64, 128, 256] Conditioning FiLM (Feature-wise Linear Modulation) Trajectory horizon H 8 waypoints Diffusion Bridge Parameters σmax _ 10.0 σmin _ 0.002 ε 0.5 Prediction target v-prediction ODE solver Heun (2nd order) Timestep schedule Karras (ρ=7.0ρ=7.0) Sampling steps k 3 (default) Training Optimizer AdamW Learning rate 1×10−41× 10^-4 Batch size 256 Epochs 30 Time sampling Continuous uniform (σmin,σmax)U( _ , _ ) Loss weighting Uniform C.2 Simulation Environments Custom Indoor. A Gazebo-based environment with multi-room layouts, narrow doorways (width ≥0.8≥ 0.8 m), and furniture obstacles. The map consists of 10 interconnected rooms over a 20×1520× 15 m area. CitySim (Outdoor). An urban environment with buildings, trees, and street-level obstacles. Navigation episodes require crossing intersections and following streets over distances of 15–30 m. Both environments use a differential-drive TurtleBot3 robot with monocular RGB camera (96×9696× 96 resolution). Ground truth paths are generated using A* search with wall-clearance cost, followed by Gaussian smoothing (σ=3.0σ=3.0). On the role of generative modeling. Deterministic regression is insufficient even when each simulated episode contains a single A*-planned path. We note two key points. First, the five real-world datasets (HuRoN, Recon, SACSoN, SCAND, GoStanford) contain human demonstrations that are inherently multi-modal: different human operators choose distinct routes through the same environment. Second, even in simulation, the conditional action distribution p(0∣,Ig)p(a_0 ,I_g) is multi-modal because many visually similar observations admit multiple valid continuations, such as turning left or right around an obstacle. Deterministic regressors average over these modes, producing infeasible split-the-difference trajectories—a well-known failure mode of behavioral cloning Florence et al. [2022], Shafiullah et al. [2022]. Table 1 confirms this empirically: ViNT (deterministic, NFE=1) achieves only 0.720 CosSim and 28% success rate, substantially below all generative methods. The motivation for RSBM is therefore not to replace deterministic planners, but to retain the multi-modal fidelity of generative policies while matching their inference cost. C.3 Statistical Variability Table 5 reports the mean and standard deviation across 3 random seeds for all methods on the Custom Indoor environment. RSBM exhibits the lowest variance across all metrics at k=3k\!=\!3, indicating particularly stable training dynamics. The high variance of NaviBridger at k=3k\!=\!3 (±0.45± 0.45 MSE, ±4%± 4\% success) further evidences the instability of unrectified bridges under few-step constraints. Table 5: Statistical variability. Mean ± std over 3 random seeds on Custom Indoor. Method k MSE↓ CosSim↑ Suc.%↑ ViNT (1-shot) 1 6.50±0.156.50± 0.15 0.720±0.0050.720± 0.005 28±228± 2 NoMaD 20 3.60±0.123.60± 0.12 0.820±0.0040.820± 0.004 32±332± 3 DDPM 50 3.80±0.143.80± 0.14 0.820±0.0050.820± 0.005 64±364± 3 FM 10 2.80±0.102.80± 0.10 0.910±0.0030.910± 0.003 82±282± 2 NaviBridger (k=10) 10 1.82±0.081.82± 0.08 0.942±0.0030.942± 0.003 88±288± 2 NaviBridger (k=3) 3 12.00±0.4512.00± 0.45 0.710±0.0150.710± 0.015 35±435± 4 RSBM (k=3) 3 1.90±0.061.90± 0.06 0.945±0.0020.945± 0.002 ±92± 2 RSBM (k=10) 10 1.72±0.051.72± 0.05 0.949±0.0020.949± 0.002 93±193± 1 C.4 Per-Dataset Multi-Metric Results Table 6 extends the main-text per-dataset comparison (Table 2) to include CosSim and FDE for all five real-world datasets, and adds DDPM and FM baselines at k=3k\!=\!3 for completeness. RSBM at k=3k\!=\!3 achieves the best or near-best performance across all metrics and all datasets, confirming the robustness of ε -rectification across diverse navigation domains ranging from small-scale indoor (SCAND) to long-range outdoor (GoStanford). Table 6: Per-dataset multi-metric comparison. All diffusion methods evaluated at k=3k\!=\!3 except NaviBridger which is also shown at its default k=10k\!=\!10. Bold: best per column. Method Metric HuRoN Recon SACSoN SCAND GoStanford Avg DDPM k=3 MSE↓ 5.20 4.80 7.10 3.40 12.50 6.60 CosSim↑ 0.520 0.480 0.380 0.590 0.310 0.456 FDE↓ 3.10 2.95 4.50 2.20 7.80 4.11 FM k=3 MSE↓ 1.85 1.52 2.30 1.10 5.40 2.43 CosSim↑ 0.820 0.790 0.720 0.850 0.640 0.764 FDE↓ 1.20 1.05 1.65 0.82 3.50 1.64 NaviBridger k=10 MSE↓ 0.27 0.88 1.43 0.59 3.22 1.28 CosSim↑ 0.955 0.935 0.920 0.960 0.875 0.929 FDE↓ 0.35 0.68 0.95 0.42 1.80 0.84 NaviBridger k=3 MSE↓ 2.80 3.45 5.20 2.15 8.50 4.42 CosSim↑ 0.750 0.690 0.620 0.780 0.520 0.672 FDE↓ 1.85 2.30 3.40 1.60 5.20 2.87 RSBM k=3 MSE↓ 0.25 0.82 1.35 0.48 3.05 1.19 CosSim↑ 0.958 0.940 0.925 0.965 0.880 0.934 FDE↓ 0.32 0.62 0.88 0.38 1.72 0.78 C.5 ODE Solver Ablation Table 7 compares Euler (1st-order) and Heun (2nd-order) solvers under RSBM (ε=0.5 =0.5) on Custom Indoor. While Heun is our default, RSBM also works well with the simpler Euler solver. Notably, Heun at k=3k\!=\!3 (NFE=5) outperforms Euler at k=5k\!=\!5 (NFE=5) at the same NFE budget, confirming the benefit of higher-order integration. Even Euler at k=5k\!=\!5 already surpasses NaviBridger’s k=10k\!=\!10 performance (MSE 1.82, CosSim 0.942), demonstrating that RSBM’s advantage stems from the rectified bridge rather than solver choice. Table 7: Solver ablation (RSBM, ε=0.5 \!=\!0.5, Custom Indoor). Heun at k=3k\!=\!3 (NFE=5) yields better results than Euler at k=5k\!=\!5 (NFE=5) at the same computational budget, confirming the benefit of higher-order integration. For reference, NaviBridger (k=10k\!=\!10) achieves MSE=1.82, CosSim=0.942. Solver k NFE MSE↓ CosSim↑ FDE↓ Suc.%↑ Euler (1st order) 3 3 2.45 0.928 1.15 82 Euler 5 5 2.05 0.938 0.95 86 Euler 10 10 1.80 0.947 0.82 90 Heun (2nd order) 3 5 1.90 0.945 0.80 92 Heun 5 9 1.78 0.948 0.76 93 Heun 10 19 1.72 0.949 0.74 93 C.6 Prior Initialization Ablation A natural question is how much performance is attributable to the learned prior gψg_ψ versus the ε -rectified bridge itself. Table 8 disentangles these contributions on Custom Indoor. Row 1 shows the prior network gψg_ψ alone (no bridge refinement). Row 2 replaces the learned prior with isotropic Gaussian noise T∼(,σmax2)a_T (0, _ ^2I) while keeping the RSBM bridge. Row 3 is the full system. Table 8: Prior initialization ablation (Custom Indoor, k=3k\!=\!3). The learned prior alone provides a reasonable but insufficient initialization; RSBM from Gaussian noise still outperforms standard SB; the full system combines both advantages. Configuration MSE↓ CosSim↑ FDE↓ Suc.%↑ Prior gψg_ψ only (no bridge) 5.80 0.780 2.60 45 Gaussian init + RSBM (ε=0.5 \!=\!0.5, k=3k\!=\!3) 4.20 0.860 1.95 62 Gaussian init + Standard SB (ε=1.0 \!=\!1.0, k=3k\!=\!3) 12.00 0.710 4.20 35 Prior gψg_ψ + Standard SB (ε=1.0 \!=\!1.0, k=3k\!=\!3) 5.50 0.810 2.35 52 Prior gψg_ψ + RSBM (ε=0.5 \!=\!0.5, k=3k\!=\!3) 1.90 0.945 0.80 92 Four key observations emerge: (i) The prior alone is insufficient. gψg_ψ without any bridge yields MSE=5.80 and only 45% success—far below the full system (MSE=1.90, 92%), confirming that RSBM contributes a 3.1×3.1× MSE reduction, not trivial smoothing. (i) Standard SB fails at low k—by design. Adding a standard Brownian Bridge (ε=1 \!=\!1) to the prior barely improves results (MSE 5.80 → 5.50). This is expected: with only 3 steps, the high-variance stochastic paths of standard SB cannot converge and instead inject noise. This is precisely the failure mode that motivates ε -rectification. (i) RSBM is effective even without a strong prior. From Gaussian noise, RSBM (ε=0.5 \!=\!0.5, k=3k\!=\!3) achieves MSE=4.20—2.9×2.9× better than standard SB from the same Gaussian initialization (MSE=12.00). This isolates the contribution of ε -rectification from prior quality. (iv) Prior and rectified bridge are complementary. The prior shortens the transport distance (MSE 12.00 → 5.80), and RSBM concentrates the remaining refinement along low-curvature paths (MSE 5.80 → 1.90). Neither component alone approaches the full system’s quality; their combination is synergistic, not redundant. C.7 Inference Cost Analysis The end-to-end inference pipeline comprises three stages: (1) vision encoding (fϕf_φ, run once per decision step), (2) prior generation (gψg_ψ, run once), and (3) iterative bridge sampling (θv_θ, run NFE times). Stages (1) and (2) constitute a fixed overhead shared by all bridge-based methods; the variable cost is entirely determined by NFE. Since RSBM and standard SB use the identical velocity network architecture and per-step computation, the 3.8×3.8× NFE reduction (from 19 to 5) translates directly into a 3.8×3.8× wall-clock speedup of the bridge sampling stage, independent of hardware. In our real-robot trials (§4.7), RSBM achieves ∼50 50 ms per decision cycle on an NVIDIA Jetson Orin, comfortably meeting the 4 Hz control rate, while DDPM requires ∼350 350 ms and fails to maintain real-time control.